Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge–Kutta time discretizations

Highly Efficient Strong Stability-Preserving Runge–Kutta Methods with Low-Storage Implementations

“…Some previous studies using SSP Runge-Kutta methods with a few extra stages have demonstrated their usefulness [17,20].…”

Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods

“…This means that conditions for the linear stability of RKDG methods are not characterized by SSP coefficients (i.e., the main theorem of SSP methods does not apply, see Theorem 2.1 of [8]) but rather by the stability region of the RK methods and the (scaled) spectral radius of the DG spatial operator. For all existing SSPRK methods, the conditions for linear stability of RKDG methods are stricter than those for nonlinear stability (see Table 1 of Section 3), and as demonstrated numerically in [22], and later in this work (see Table 5 of Section 5), it is the linear stability conditions that must be respected in practice or the high-order convergence of the RKDG methods will degenerate to first-order. The main consequence of this is that previously derived SSPRK methods with optimal SSP coefficients are not optimal for DG spatial discretizations, that is, they do not maximize the allowable time step size that will maintain both higher order convergence and stability.…”

Time step restrictions for Runge–Kutta discontinuous Galerkin methods on triangular grids

“…For an RKDG method of a given order, the computational efficiency is a function of both the CFL condition and the number of stages of the SSP RK method. In the majority of cases examined in [15], use of the s > k SSP RK methods was more efficient than the standard practice of using the s ¼ k methods. This was due to gains in allowable time step size afforded by improved CFL conditions for linear stability that were large enough to offset the additional work introduced by the increased number of stages.…”

“…This condition is, in fact, exact for the cases p ¼ 0 and p ¼ 1 (the case p ¼ 0 can be proven trivially and the case p ¼ 1 was proven in [7] for p P 2, this estimate was observed to be less than 5% smaller than numerically-obtained estimates of the CFL condition [9]. Given the somewhat restrictive nature of (1), especially as p increases, the stability of RKDG methods using so-called strong-stability-preserving (SSP) RK methods where the number of stages s is greater than the order k of the method were examined in [15] with the goal of obtaining more efficient RKDG methods. Necessary time step restrictions for stability were derived and ''optimal" RKDG methods, in terms of computationally efficiency, were identified.…”

“…In recent numerical experiments, we have noted that simple two-dimensional analogs of these CFL conditions do not appear to be sufficient for stability in two-dimensions on structured triangular grids. Therefore, in this paper, we seek to extend the types of one-dimensional analyses performed in [7,9,15] to RKDG methods in two-dimensions. Specifically, we derive necessary time step restrictions, which also appear to be sufficient, for the stability of RKDG methods on structured triangular grids applied to the constant coefficient, twodimensional transport equation with periodic boundary conditions.…”